Question

Find the area of the region bounded by the curve $y^2=4xand{x}^2=4y$.

Answer 1

Given equation of curves are

$y^2=4x$ ........(i)

and $x^2=4y$ ........(ii)

$\Rightarrow {\left(\frac{x^2}{4}\right)}^2=4x\Rightarrow \frac{x^4}{4.4}=4x\Rightarrow x^4=64x\Rightarrow x^4-64x=0\Rightarrow x(x^3-4^3)=0\Rightarrow x=4,0$

$\therefore$ Area of shaded region, $A=A={\int }_0^4(\sqrt{4}x-\frac{x^2}{4})dx={\int }_4^0(2\sqrt{x}-\frac{x^2}{4})dx={[\frac{2x^{\frac{3}{2}}.2}{3}-\frac{1}{4}.\frac{1}{4}.\frac{x^3}{3}]}_0^4=\frac{2.2}{3}.8-\frac{1}{4}.\frac{64}{3}-0=\frac{32}{3}-\frac{16}{3}=\frac{16}{3}sq.units$